Journal of Applied Electrochemistry

, Volume 37, Issue 5, pp 605–616 | Cite as

Adaptive, multi-parameter battery state estimator with optimized time-weighting factors

  • Mark Verbrugge
Original Paper


We derive and implement a battery control algorithm that can accommodate an arbitrary number of model parameters, with each model parameter having its own time-weighting factor, and we propose a method to determine optimal values for the time-weighting factors. Time-weighting factors are employed to give greater impact to recent data for the determination of a system’s state. We employ the (controls) methodology of weighted recursive least squares, and the time weighting corresponds to the exponential-forgetting formalism. The output from the adaptive algorithm is the battery state of charge (remaining energy), state of health (relative to the battery’s nominal performance), and predicted power capability. Results are presented for a high-power lithium ion battery.


Battery Control Equivalent circuit Mathematical model Power prediction State of charge prediction Weighted recursive least squares 

List of symbols


Coulombic capacity, C-h/s


1/C D , 1/F


Regressed intercept


1/ (R ct C D ), 1/s


Capacitance, F


Current, A


Number of parameters m to be regressed adaptively


Parameter to be regressed


Number of time steps (data points) in the regression


Power, W


Capacitance ratio,C D,discharge/C D,charge


High-frequency resistance, ohm


Effective interfacial resistance, ohm


Percent state of charge (energy content in the battery relative to the energy content upon full charge)


State of health, Eq. 15




Time, s


System voltage, V


Hysteresis voltage, V


Open-circuit voltage, V


Time-dependent values multiplying onto parameters m


Dependent variable


Weighting factor (Eq. 17)


Hysteresis parameter, C−1


Error or loss term


Unweighted total error as defined by Eq. 13


Forgetting factor (Eq. 2)


Parameter for selective weighting of data beyond that of the forgetting factor


Current efficiency





The authors recognize and thank Ramona Ying of General Motors R&D (Chemical and Environmental Sciences Laboratory) for acquiring the lithium ion battery data and helpful discussions with Damon Frisch and Brian Koch of the GM Electrical Center as well as Mutasim Salman of the Electrical Controls and Integration Laboratory at GM R&D.


  1. 1.
    Pillar S, Perrin M, Jossen A (2001) J Power Sources 96:113CrossRefGoogle Scholar
  2. 2.
    Verbrugge MW, Tate ED (2004) J Power Sources 126:236CrossRefGoogle Scholar
  3. 3.
    Verbrugge MW, Liu P, Soukiazian S (2005) J Power Sources 141:369CrossRefGoogle Scholar
  4. 4.
    Verbrugge MW, Frisch D, Koch B (2005) J Electrochem Soc 152:A333CrossRefGoogle Scholar
  5. 5.
    Verbrugge MW, Koch BJ (2006) J Electrochem Soc 153:A187CrossRefGoogle Scholar
  6. 6.
    Gelb A (ed) (1974) Applied optimal estimation. MIT Press, Cambridge, MAGoogle Scholar
  7. 7.
    Anderson BDO, Moore JB (1979) Optimal filtering. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  8. 8.
    Maybeck PS (1979) Stochastic models, estimation and control, vol. 141–1 of Mathematics in science and engineering, Academic PressGoogle Scholar
  9. 9.
    Widrow B, Stearn SD (1985) Adaptive signal processing. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  10. 10.
    Brogan WL (1985) Modern control theory, 2nd edn. Prentice-Hall, Englewood Cliffs NJGoogle Scholar
  11. 11.
    Ljnug L, Söderström T (1986) Theory and practice of recursive identification, MIT PressGoogle Scholar
  12. 12.
    Bellanger MG (1987) Adaptive digital filters and signal analysis. Marcel Dekker, New York, NYGoogle Scholar
  13. 13.
    Åström KJ, Wittenmark B (1989) Adaptive control, Addison-WesleyGoogle Scholar
  14. 14.
    Kulhavý R (1996) Recursive nonlinear estimation. A geometric approach. Springer, LondonGoogle Scholar
  15. 15.
    Fortescue TR, Kershenbaum LS, Ydstie BE (1981) Automatica 17:831CrossRefGoogle Scholar
  16. 16.
    Bittanti S, Bolzern P, Campi M, Coletti E (1988) Proceedings of the American Control Conference, IEEE, Austin, Texas, pp 1530–1531, December 1988Google Scholar
  17. 17.
    Ljung L, Gunnarsson S (1990) Automatica 26:7CrossRefGoogle Scholar
  18. 18.
    Parkum JE, Poulsen NK, Holst J (1992) Int J Control 55:109CrossRefGoogle Scholar
  19. 19.
    Kulhavý R (1993) Int J Control 58:905CrossRefGoogle Scholar
  20. 20.
    Vahidi A, Druzhinina M, Stefanopoulou A, Peng H (2003) Proceedings of the American Control Conference, IEEE, Denver, Colorado, pp 4951–4956, June 2003Google Scholar
  21. 21.
    Zheng Y, Lin Z (2003) IEEE transactions on circuits and systems—II: analogue and digital signal processing 50:602CrossRefGoogle Scholar
  22. 22.
    Berndt D (1993) Maintenance-Free Batteries. Research Studies Press, Taunton, Somerset, UKGoogle Scholar
  23. 23.
    Conte SD, de Boor C (1980) Elementary numerical analysis, 3rd edn. McGraw-Hill, New York, NYGoogle Scholar
  24. 24.
    Newman J, Tiedemann W (1975) AIChE J 21:25CrossRefGoogle Scholar
  25. 25.
    Newman J (1991) Electrochemical systems, 2nd edn. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  26. 26.
    Verbrugge MW (1995) J Electrostatics 34:61CrossRefGoogle Scholar
  27. 27.
    Verbrugge MW (1995) AIChE J 41:1550CrossRefGoogle Scholar
  28. 28.
    Baker DR, Verbrugge MW (1999) J Electrochem Soc 146:2413CrossRefGoogle Scholar
  29. 29.
    Thomas KE, Darling RM, Newman J (2002) In: van Schalkwaijk W, Scrosati B (eds) chap. 12, Advances in lithium-ion batteries. Kluwer Academic/PlenumGoogle Scholar
  30. 30.
    Beyer WH (ed) (1976) Standard mathematical tables, 24th edn. CRC Press, Cleveland OHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.General Motors Research and DevelopmentWarrenUSA

Personalised recommendations